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Metamath Proof Explorer

Theorem pm2.61iii

Description: Inference eliminating three antecedents. (Contributed by NM, 2-Jan-2002) (Proof shortened by Wolf Lammen, 22-Sep-2013)

Ref Expression
Hypotheses pm2.61iii.1 ⊢ ( ¬ 𝜑 → ( ¬ 𝜓 → ( ¬ 𝜒 → 𝜃 ) ) )
pm2.61iii.2 ⊢ ( 𝜑 → 𝜃 )
pm2.61iii.3 ⊢ ( 𝜓 → 𝜃 )
pm2.61iii.4 ⊢ ( 𝜒 → 𝜃 )
Assertion pm2.61iii 𝜃

Proof

Step Hyp Ref Expression
1 pm2.61iii.1 ⊢ ( ¬ 𝜑 → ( ¬ 𝜓 → ( ¬ 𝜒 → 𝜃 ) ) )
2 pm2.61iii.2 ⊢ ( 𝜑 → 𝜃 )
3 pm2.61iii.3 ⊢ ( 𝜓 → 𝜃 )
4 pm2.61iii.4 ⊢ ( 𝜒 → 𝜃 )
5 2 a1d ⊢ ( 𝜑 → ( ¬ 𝜒 → 𝜃 ) )
6 3 a1d ⊢ ( 𝜓 → ( ¬ 𝜒 → 𝜃 ) )
7 1 5 6 pm2.61ii ⊢ ( ¬ 𝜒 → 𝜃 )
8 4 7 pm2.61i ⊢ 𝜃